Errors in the arguments contradicting a question present in Rudin's Principles of Mathematical Analysis – math.stackexchange.com

In page 43 of Rudin's Principles of Mathematical Analysis, there's a question

Let $A_{1}, A_{2}, A_{3}, \dots, A_{n}$ be subsets of a metric space. If $B_{n} = \bigcup_{i = 1}^{n} A_{i}$, then $...